Mathematics for the interested outsider

A few more groups

I want to throw out a few more examples of groups before I move deeper into the theory.

First up: Abelian groups. These are more a general class of groups than an example like permutation groups were. They are distinguished by the fact that the composition is “commutative” — it doesn’t matter what order the group elements come in. The composition is the same as the composition .

All the groups I’ve mentioned so far, except for permutations and rotations, are Abelian. It’s common when dealing with an abelian group to write the composition as “+”, the identity as “0”, and the inverse of as . Let’s recap the axioms for an Abelian group in this notation.

For any , , and :

There is an element 0 so that for any :

For every there is an element so that:

For any and :

Abelian groups are really fantastically important. Many later algebraic structures start with an Abelian group and add structure to it, just as a group starts with a set and adds structure to it. We’ll see many examples of these later.

The other thing I want to mention is a free group. As the name might imply, this is a group with absolutely no restrictions other than the group axioms. We start by picking some basic pieces, sometimes called “generators” or “letters”, and then just start writing out whatever “words” the rules of group theory allow.

Let’s start with the free group on one letter: . We definitely have the identity element — written “1” — and we throw in our single letter . We can compose this with itself however many times we like by just writing letters next to each other: , , , and so on. We also need an inverse, . We can use and to build up long words like . But notice that whenever an and an sit next to each other they cancel. That collapses this long word down to . We see that in all words look like , where a positive means a string of s, a negative means a string of s, and for the identity. We compose just by adding the exponents.

The free group on two letters, gets a lot more complicated. We again start with the identity and throw in letters and . Now we can build up all sorts of words like . But now we can’t do anything to pull and past each other. Letters only cancel their inverses when they’re right next to each other, so this word only collapses to . That’s the best we can do. Free groups on more generators are pretty much the same, but with more basic symbols.

Composition of words and just writes them one after another, cancelling whatever possible in the middle. For example, in let’s say and . We write them together (in order!) as and cancel inverses where we can to get .

Free groups seem hideously complicated at first, but they aren’t so bad once you get used to them. They’re also tremendously useful, as we’ll soon see. They’re the primordial groups, with absolutely nothing extra beyond the bare minimum of what’s needed to make a group.

A group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses (disregarding trivial variations such as ). Apart from the existence of inverses no other relation exists between the generators of a free group.——wikipedia

A group is called a free group if no relation exists between its group generators other than the relationship between an element and its inverse required as one of the defining properties of a group. ——wolfram mathworld

Sort of. The elements of a free group are all the finite sequences of generators or their inverses, and words are composed by concatenation of the sequences. The empty sequence is the identity (which you should easily verify), and any word has an inverse (as the post asks you to verify). Does this seem clearer?

Doesn’t the identity have to exist as an element of the group? If so, the elements of a free group consist of all finite sequences of generators or their inverses or the empty sequence, and words are composed by the concatenation of sequences.

Indeed it would be useful. Group actions and symmetries are definitely on the agenda. I just want to have homomorphisms under my belt so I can do them neatly, and for that I’ll need functions. Stay tuned.

[…] Ab-Categories Now that we’ve done a whole lot about enriched categories in the abstract, let’s look at the very useful special case of categories enriched over — the category of abelian groups. […]

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